Math: Conic Sections: Hyperbolas and Parabolas Worksheet

Question 1

For the parabola (x - 2)^2 = 8(y + 1), find the vertex, focus, directrix, and direction of opening.

Accepted Answer

The vertex is (2, -1). Since 4p = 8, p = 2, so the focus is (2, 1), the directrix is y = -3, and the parabola opens upward.

Question 2

Write the equation of the parabola with vertex (-3, 4) and focus (-3, 1).

Accepted Answer

The focus is 3 units below the vertex, so p = -3. The equation is (x + 3)^2 = -12(y - 4).

Question 3

Rewrite y = x^2 - 6x + 11 in vertex form. Then state the vertex and direction of opening.

Accepted Answer

The equation in vertex form is y = (x - 3)^2 + 2. The vertex is (3, 2), and the parabola opens upward because the coefficient of the squared term is positive.

Question 4

Write the equation of the parabola with focus (5, 2) and directrix x = 1.

Accepted Answer

The vertex is halfway between the focus and directrix at (3, 2). Since p = 2, the equation is (y - 2)^2 = 8(x - 3).

Question 5

For the equation y^2 + 4y = 12x - 8, put the parabola in standard form and find the vertex, focus, and directrix.

Accepted Answer

Completing the square gives (y + 2)^2 = 12(x - 1/3). The vertex is (1/3, -2), p = 3, the focus is (10/3, -2), and the directrix is x = -8/3.

Question 6

For the hyperbola (x - 1)^2/9 - (y + 2)^2/16 = 1, find the center, vertices, foci, and equations of the asymptotes.

Accepted Answer

The center is (1, -2). Since a = 3, b = 4, and c = 5, the vertices are (-2, -2) and (4, -2), the foci are (-4, -2) and (6, -2), and the asymptotes are y + 2 = (4/3)(x - 1) and y + 2 = -(4/3)(x - 1).

Question 7

Write the equation of the hyperbola with center (0, 0), vertices (0, 5) and (0, -5), and foci (0, 13) and (0, -13).

Accepted Answer

The hyperbola opens up and down, so its form is y^2/a^2 - x^2/b^2 = 1. Since a = 5 and c = 13, b^2 = 13^2 - 5^2 = 144, so the equation is y^2/25 - x^2/144 = 1.

Question 8

Find the equations of the asymptotes for the hyperbola (y - 3)^2/4 - (x + 1)^2/25 = 1.

Accepted Answer

The center is (-1, 3), a = 2, and b = 5. Because the hyperbola is vertical, the asymptotes are y - 3 = (2/5)(x + 1) and y - 3 = -(2/5)(x + 1).

Question 9

Put 9x^2 - 16y^2 - 54x - 64y - 127 = 0 in standard form. Then state the center and transverse axis direction.

Accepted Answer

Completing the square gives 9(x - 3)^2 - 16(y + 2)^2 = 144, so the standard form is (x - 3)^2/16 - (y + 2)^2/9 = 1. The center is (3, -2), and the transverse axis is horizontal.

Question 10

Classify x^2 - 4x - 8y + 20 = 0 as a parabola or hyperbola. Then write it in standard form and state its vertex.

Accepted Answer

This equation represents a parabola because only one variable is squared. Completing the square gives (x - 2)^2 = 8(y - 2), so the vertex is (2, 2).

Question 11

A hyperbola has foci (-4, 0) and (4, 0). For every point on the hyperbola, the absolute difference of the distances to the foci is 6. Write the equation in standard form.

Accepted Answer

The center is (0, 0), c = 4, and 2a = 6, so a = 3. Since b^2 = c^2 - a^2 = 16 - 9 = 7, the equation is x^2/9 - y^2/7 = 1.

Question 12

A parabolic reflector has cross-section y = (1/12)x^2 with vertex at the origin. Find the focus of the parabola.

Accepted Answer

Rewrite the equation as x^2 = 12y. Since 4p = 12, p = 3, so the focus is (0, 3).

Math: Conic Sections: Hyperbolas and Parabolas

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